Spectrum Analysis

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Spectrum Analysis1 Population SpectrumA random variable Yt, which follows weak stationary process, can be represented as a weighted sum ofcos(!t) and sin(!t), letting! be a certain frequency. The form of this represantation isYt = +Z 0(!)cos(!t)d! +Z 0(!)sin(!t)d!The basic idea of spectral analysis is to measure how important diﬀerent frequencies are to explain thechanging of Yt. All of the weak stationary process has the two form of representation; frequency domainrepresantation and time domain representation, which do not exclude one other.LetfYtg1t= 1 be weak stationary process with mean and jth autocovariancej , wherej is absolutelysummable. Then deﬁne the autocovariance generating functiongY(z) to begY(z)def=1Xj = 1j zjIn the equation above, lettingzj = ei! and dividing by 2 , we obtainSY(!) =12gY(ei! ) =121Xj = 1j ei!j (1)Here notice that a spectrum is a function of!, therefore giving a value of! and fj g1j = 1 ,SY(!) can becalculated. According to De Moivre’s theorem, which states thatei!j = cos(!j ) sin(!j )and substituing thie equation into Eq.(1), we getSY(!) =121Xj = 1jcos(!j ) i sin(!j )=120(cos0 i sin0) +121Xj =1cos(!j ) + cos( !j ) i sin(!j ) i sin( !j )=12h0 + 21Xj =1j cos(!j )i(2)where the second equality holds from the fact thatj = j in a weak stationary process. Iffj g1j = 1 isabsolutely summable then there exists population spectrum by Eq.(2) andSY(!) is a continuous functionof!. Als

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Spectrum Analysis1 Population SpectrumA random variable Yt, which follows weak stationary process, can be represented as a weighted sum ofcos(!t) and sin(!t), letting! be a certain frequency. The form of this represantation isYt = +Z 0(!)cos(!t)d! +Z 0(!)sin(!t)d!The basic idea of spectral analysis is to measure how important diﬀerent frequencies are to explain thechanging of Yt. All of the weak stationary process has the two form of representation; frequency domainrepresantation an..